Based on Ostrowski's method, a new family of eighthorder iterative methods for solving nonlinear equations by using weight function methods is presented. Per iteration the new methods require
three evaluations of the function and one evaluation of its first derivative. Therefore, this family of methods has the efficiency index which equals 1.682. Kung and Traub conjectured that a multipoint iteration without
memory based on
In this paper, we consider iterative methods to find a simple root
To improve the local order of convergence, many modified methods have been proposed in the open literature; see [
In this paper, based on Ostrowski’s method, we present a new family of optimal eighthorder methods by using the method of weight functions and we apply a few weight functions to construct families of iterative methods with high convergence and high efficiency index. The convergence analysis is provided to establish their eighth order of convergence. In terms of computational cost, they require the evaluations of only three functions and one firstorder derivative per iteration. This gives 1.682 as efficiency index of the presented methods. The new methods are comparable with Bi et al.’s method, Liu and Wang’s method, and Sharma’s method. The efficacy of the methods is tested on a number of numerical examples.
We use the symbols
In order to construct new methods, we consider an iteration scheme of the form
Assume that
According to the above analysis, we have proved the following theorem.
Assume that
Any method of the family (
In what follows, we give some concrete iterative forms of scheme (
The functions
The functions
The functions
Now, Method
The examples considered in this study.
Test functions  Zeros 

















Numerical computations reported here have been carried out in a
Comparison of various iterative methods under the same total number of function evaluations (TNFE = 12).
( 
( 
( 
(BM)  (LWM)  (ShM)  

















COC  8.000000  8.000000  8.000000  8.000000  8.000000  8.000000 


















COC  8.000001  8.000002  8.000002  7.999999  8.000000  8.000000 


















COC  7.986895  7.993900  7.984913  7.965999  7.899280  7.991216 


















COC  7.999999  7.999997  8.000018  7.999999  7.999999  8.000000 


















COC  8.000000  8.000000  8.000000  8.000000  8.000000  8.000000 


















COC  8.000000  8.000000  8.000000  8.000000  8.000000  8.000000 


















COC  8.015338  8.000253  8.000189  8.000058  8.000030  7.999924 


















COC  8.000000  8.000000  8.000000  8.000000  8.000000  8.000000 
We have obtained a new family of variants of Ostrowski’s method. The convergence order of these methods is eight, which consist of three evaluations of the function and one evaluation of the first derivative per iteration, so they have an efficiency index equal to
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors are grateful to the referees for their comments and suggestions that helped to improve the paper. This research was supported by Islamic Azad University, Hamedan Branch.